Weingast B.R., Wittman D. The Oxford Handbook of Political Economy

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422 interpersonal comparisons of well-being

theorem is to consider informational environments that allow for interpersonal com-

parisons of well-being.

We focus on establishing a social ranking of alternatives in this chapter. Because

actual decision problems typically are choice problems, it is natural to ask what to do

about constraints facing a society, such as those resulting from resource limitations.

The approach to constrained social choice implicitly followed in this chapter is that

of constrained optimization. We propose to select, for each choice problem identiﬁed

by a feasible set of alternatives, a best element according to the social objective

represented by the social ordering. If the suitability of a social objective on ethical

grounds is independent of the constraints (a position we advocate), our approach

which does not model constraints explicitly does not involve any loss of generality.

The model discussed in this chapter can be generalized in various ways. Two of the

most important extensions involve considerations of uncertainty and the possibility

of population change. Due to space constraints, we cannot present them in detail

and provide brief summaries instead, accompanied by some suggestions for further

reading.

Welfarist social evaluation under uncertainty is discussed, for example, in Black-

orby, Bossert, and Donaldson (2002, 2003), Blackorby, Donaldson, and Weymark

(1999, 2006), Harsanyi (1955, 1977), Mongin (1994), and Weymark (1991, 1993, 1994,

1995). While most contributions (such as those of Harsanyi) focus on the ranking

of probability distributions, an attractive alternative is to ﬁx the probabilities of the

possible states that may occur and represent the social choice situation under uncer-

tainty by considering prospects—vectors of alternatives, one for each possible state

that may materialize. This model is a natural generalization of the one considered

here. Instead of actual (ex post) utility functions, ex ante utilities are employed and a

welfarism theorem that is formulated in terms of ex ante utility functions is obtained.

See Blackorby, Bossert, and Donaldson (2003) for details.

Variable population social choice is discussed, for example, in Blackorby and

Donaldson (1984) and in Blackorby, Bossert, and Donaldson (2005b). As is the

case for uncertainty, natural generalizations of welfarism can be characterized if the

population is allowed to vary from one alternative to another. Moreover, the model

can be extended to cover both uncertainty and variable population issues at the

same time.

References

Arrow,K.1951. Social Choice and Individual Values.NewYork:Wiley(2nd edn. 1963).

Blackorby,C.,Bossert,W.,andDonaldson,D.2002. Utilitarianism and the theory of

justice. Pp. 543–96 in The Handbook of Social Choice and Welfare,vol.i,ed.K.Arrow,A.Sen,

and K. Suzumura. Amsterdam: Elsevier.

2003. Harsanyi’s theorem: a multi-proﬁle approach and variable-population

extensions. Discussion Paper 03–2003, Université de Montréal, CIREQ.

charles blackorby & walter bossert 423

2005a. Multi-proﬁle welfarism: a generalization. Social Choice and Welfare,

24: 253–67.

2005b. Population Issues in Social-Choice Theory, Welfare Economics and Ethics.

Cambridge: Cambridge University Press.

and Donaldson,D.1982. Ratio-scale and translation-scale full interpersonal compa-

rability without domain restrictions: admissible social evaluation functions. International

Economic Review, 23: 249–68.

1984. Social criteria for evaluating population change. Journal of Public Economics,

25: 13–33.

1992. Pigs and guinea pigs: a note on the ethics of animal exploitation. Economic

Journal, 102: 1345–69.

and Weymark,J.1984. Social choice with interpersonal utility comparisons: a

diagrammatic introduction. International Economic Review, 25: 327–56.

1999. Harsanyi’s social aggregation theorem for state-contingent alternatives.

Journal of Mathematical Economics, 32: 365–87.

2006. Social aggregation and the expected utility hypothesis. In Justice, Po-

litical Liberalism, and Utilitarianism: Themes from Harsanyi and Rawls,ed.M.Sallesand

J. Weymark. Cambridge: Cambridge University Press, forthcoming.

Blackwell,D.,andGirshick,M.1954. Theory of Games and Statistical Decisions.NewYork:

Wiley.

Blau,J.1957. The existence of social welfare functions. Econometrica, 25: 302–13.

Bossert,W.,andWeymark,J.2004.Utilityinsocialchoice.Pp.1099–177 in The Handbook

of Utility Theory, vol. ii: Extensions, ed. S. Barberà, P. Hammond, and C. Seidl. Dordrecht:

Kluwer.

d’Aspremont,C.1985. Axioms for social welfare orderings. Pp. 19–76 in Social Goals and

Social Organizations: Essays in Memory of Elisha Pazner, ed. L. Hurwicz, D. Schmeidler, and

H. Sonnenschein. Cambridge: Cambridge University Press.

and Gevers,L.1977. Equity and the informational basis of collective choice. Review of

Economic Studies, 44: 199–209.

2002. Social welfare functionals and interpersonal comparability. Pp. 459–541 in

The Handbook of Social Choice and Welfare, vol. i, ed. K. Arrow, A. Sen, and K. Suzumura.

Amsterdam: Elsevier.

DeMeyer,F.,andPlott,C.1971. A welfare function using “relative intensity” of preference.

Quarterly Journal of Economics, 85: 179–86.

Deschamps,R.,andGevers,L.1978. Leximin and utilitarian rules: a joint characterization.

Journal of Economic Theory, 17: 143–63.

Dixit,A.1980. Interpersonal comparisons and social welfare functions. Unpublished manu-

script, University of Warwick, Dept. of Economics.

Gevers,L.1979. On interpersonal comparability and social welfare orderings. Econometrica,

47: 75–89.

Goodin,R.1991. Utility and the good. Pp. 241–8 in A Companion to Ethics, ed. P. Singer.

Oxford: Basil Blackwell.

Griffin,J.1986. Well-Being: Its Meaning, Measurement, and Moral Importance.Oxford:

Clarendon Press.

Hammond,P.1976. Equity, Arrow’s conditions, and Rawls’ diﬀerence principle. Econometrica,

44: 793–804.

1979. Equity in two person situations: some consequences. Econometrica, 47: 1127–35.

Harsanyi,J.1955. Cardinal welfare, individualistic ethics, and interpersonal comparisons of

utility. Journal of Political Economy, 63: 309–21.

424 interpersonal comparisons of well-being

Harsanyi,J.1977. Rational Behavior and Bargaining Equilibrium in Games and Social Situa-

tions. Cambridge: Cambridge University Press.

Maskin,E.1978. A theorem on utilitarianism. Review of Economic Studies, 45: 93–6.

Milnor,J.1954. Games against nature. Pp. 49–59 in Decision Processes,ed.R.Thrall,

C. Coombs, and R. Davis. New York: Wiley.

Mongin,P.1994. Harsanyi’s aggregation theorem: multi-proﬁle version and unsettled ques-

tions. Social Choice and Welfare, 11: 331–54.

Roberts,K.1980a. Possibility theorems with interpersonally comparable welfare levels. Re-

view of Economic Studies, 47: 409–20.

1980b. Interpersonal comparability and social choice theory. Review of Economic Studies,

47: 421–39.

Sen,A.1970. Collective Choice and Social Welfare. San Francisco: Holden-Day.

1974. Informational bases of alternative welfare approaches: aggregation and income

distribution. Journal of Public Economics, 3: 387–403.

1977. On weights and measures: informational constraints in social welfare analysis.

Econometrica, 45: 1539–72.

1979. Personal utilities and public judgements: or what’s wrong with welfare economics?.

Economic Journal, 89: 537–58.

1986. Social choice theory. Pp. 1073–181 in The Handbook of Mathematical Economics,vol.

iii, ed. K. Arrow and M. Intriligator. Amsterdam: North-Holland.

Suppes,P.1966. Some formal models of grading principles. Synthese, 6: 284–306.

Weymark,J.1991. A reconsideration of the Harsanyi–Sen debate on utilitarianism.

Pp. 255–320 in Interpersonal Comparisons of Well-Being, ed. J. Elster and J. Roemer.

Cambridge: Cambridge University Press.

1993. Harsanyi’s social aggregation theorem and the weak Pareto principle. Social Choice

and Welfare, 10: 209–22.

1994. Harsanyi’s social aggregation theorem with alternative Pareto principles. Pp. 869–87

in Models and Measurement of Welfare and Inequality, ed. W. Eichhorn. Berlin: Springer.

1995. Further remarks on Harsanyi’s social aggregation theorem and the weak Pareto

principle. Social Choice and Welfare, 12: 87–92.

chapter 24

..............................................................

FAIR DIVISION

..............................................................

steven j. brams

1 Introduction

.............................................................................

The literature on fair division has burgeoned in recent years, with ﬁve academic

books (Young 1994; Brams and Taylor 1996;RobertsonandWebb1998; Moulin 2003;

Barbanel 2005) and one popular book (Brams and Taylor 1999b) providing overviews.

In this review, I will give a brief survey of three diﬀerent literatures: (i) division of a

single heterogeneous good (e.g. a cake with diﬀerent ﬂavors or toppings); (ii) division,

in whole or part, of several divisible goods; and (iii) allocation of several indivisible

goods. In each case, I assume that diﬀerent people, called players,mayhavediﬀerent

preferences for the items being divided.

For (i) and (ii), I will describe and illustrate procedures for dividing divisible goods

fairly, based on diﬀerent criteria of fairness. For (iii), I will discuss problems that arise

in allocating indivisible goods, highlighting trade-oﬀs that must be made when not

all criteria of fairness can be satisﬁed simultaneously.

2 Single Heterogeneous Good

.............................................................................

The metaphor I use for a single heterogeneous good is a cake, with diﬀerent ﬂavors

or toppings, that cannot be cut into pieces that have exactly the same composition.

Unlike a sponge or layer cake, diﬀerent players may like diﬀerent pieces—even if they

are the same physical size—because they are not homogeneous.

∗

I am grateful to the editors for their valuable comments on an earlier draft.

426 fair division

Some of the cake-cutting procedures that have been proposed are discrete, whereby

players make cuts with a knife—usually in a sequence of steps—but the knife is not

allowed to move continuously over the cake. Moving-knife procedures, on the other

hand, permit such continuous movement and allow players to call “stop” at any point

at which they want to make a cut or mark.

There are now about a dozen procedures for dividing a cake among three players,

and two procedures for dividing a cake among four players, such that each player is

assured of getting a most valued or tied-for-most-valued piece (Brams, Taylor, and

Zwicker 1995; Barbanel and Brams 2004). When a cake can be so divided, no player

will envy another player, resulting in an envy-free division.

In the literature on cake-cutting, two assumptions are commonly made:

1. The goal of each player is to maximize the minimum-valued piece (maximin

piece) he or she can guarantee for himself or herself, regardless of what the other

players do. To be sure, a player might do better by not following such a maximin

strategy; this will depend on the strategy choices of the other players. However,

all players are assumed to be risk averse: They never choose strategies that might

yield them larger pieces if they entail the possibility of giving them less than their

maximin pieces.

2. The preferences of the players over the cake are continuous. Consider a proce-

dure in which a knife moves across a cake from left to right and, at any moment,

the piece of the cake to the left of the knife is A and the piece to the right is B.

The continuity assumption enables one to use the intermediate value theorem

to say the following: if, for some position of the knife, a player views piece A as

being larger than piece B, and for some other position he or she views piece B as

being larger than piece A, then there must be some intermediate position such

that the player values the two pieces exactly the same.

Only two three-person procedures (Stromquist 1980; Barbanel and Brams 2004),

and no four-person procedure, make an envy-free division with the minimal number

of cuts (n −1cutsiftherearen players). A cake so cut ensures that each player gets a

single connected piece, which is especially desirable in certain applications (e.g. land

division).

For two players, the well-known procedure of “I cut the cake, you choose a piece,”

or “cut-and-choose,” leads to an envy-free division if the players choose maximin

strategies. The cutter divides the cake 50–50 intermsofhisorherpreferences.

(Physically, the two pieces may be of diﬀerent size, but the cutter values them the

same.) The chooser takes the piece he or she values more and leaves the other piece

for the cutter (or chooses randomly if the two pieces are tied in his or her view).

Clearly, these strategies ensure that each player gets at least half the cake, as he or she

values it, proving that the division is envy free.

But this procedure does not satisfy other desirable properties. For example, if the

cake is, say, half vanilla, which the cutter values at 75 per cent, and half chocolate,

whichthechooservaluesat75 per cent, a “pure” vanilla–chocolate division would be

better for both players than the divide-and-choose division, which gives each player

steven j. brams 427

50 per cent vanilla and 50 percentchocolate,oreachexactly50 per cent of the value

of the cake.

The moving-knife equivalent of “I cut, you choose” is for a knife to move contin-

uously across the cake, say from left to right. Assume that the cake is cut when one

player calls “stop.” If each of the players calls “stop” when he or she perceives the knife

to be at a 50–50 point, then the ﬁrst player to call “stop” will produce an envy-free

division if the cake is cut at this point and the ﬁrst player gets the left piece, which

as the cutter he or she values at 50 percent,andtheotherplayergetstherightpiece,

which as the chooser he or she values more. (If both players call “stop” at the same

time, the pieces can be randomly assigned to the two players.) To be sure, if the player

who would truthfully call “stop” ﬁrst knows the other player’s preferences and delays

calling “stop” until just before the knife would reach the other player’s 50–50 point,

the ﬁrst player can obtain a greater-than-50 per cent share on the left. However, the

possession of such information by the cutter is not generally assumed in justifying

cut-and-choose, though it does not undermine an envy-free division.

Surprisingly, to go from two players making one cut to three players making two

cuts cannot be done by a discrete procedure if the division is to be envy free.

The

three-person discrete procedure that uses the fewest cuts is one discovered indepen-

dently by John L. Selfridge and John H. Conway about 1960; it is described in, among

other places, Brams and Taylor (1996) and Robertson and Webb (1998) and requires

up to ﬁve cuts.

Although there is no discrete four-person envy-free procedure that uses a bounded

number of cuts, Brams, Taylor, and Zwicker (1997) and Barbanel and Brams (2004)

give moving-knife procedures that require up to eleven and ﬁve cuts, respectively.

The Brams–Taylor–Zwicker (1997) procedure is arguably simpler because it requires

fewer simultaneously moving knives. Peterson and Su (2002) give a four-person envy-

free moving-knife procedure for chore division, whereby each player thinks he or she

receives the least undesirable chores, that requires up to sixteen cuts.

To illustrate ideas, I will describe the Barbanel–Brams (2004) three-person, two-

cut envy-free procedure, which is based on the idea of squeezing a piece by moving

two knives simultaneously.

This procedure, however, is not as complex as Brams

and Taylor’s (1995) general n-person discrete procedure. Their procedure illustrates

the price one must pay for an envy-free procedure that works for all n, because it

places no upper bound on the number of cuts that are required to produce an envy-

free division; this is also true of other n-person envy-free procedures (Robertson and

Web b 1997;Pikhurko2000). While the number of cuts needed depends on the players’

preferences over the cake, it is worth noting that Su’s (1999) approximate envy-free

¹Fordiﬀerent procedures to attenuate this and other problems, see Brams, Jones, and Klamler 2004.

²RobertsonandWebb1998, 28–9; additional information on the minimum numbers of cuts required

to give envy-freeness is given in Even and Paz 1984, and Shishido and Zeng 1999.

³ The Barbanel–Brams (2004) four-person, ﬁve-cut envy-free procedure also uses this idea, but it is

considerably more complicated and will not be described here. For recent results on pie-cutting, in

which radial cuts are made from the center of a pie to divide it into wedge-shaped pieces, see Barbanel

and Brams 2006, and Brams, Jones, and Klamler 2006b.

428 fair division

procedure uses the minimal number of cuts at the cost of only small departures from

envy-freeness.

I next describe the Barbanel–Brams three-person, two-cut envy-free procedure,

called the squeezing procedure (Barbanel and Brams 2004). I refer to players by

number—player 1,player2, and so on—calling even-numbered players “he” and odd-

numbered players “she.” Although cuts are made by two knives in the end, initially

one player makes “marks,” or virtual cuts, on the line segment deﬁning the cake; these

marks may subsequently be changed by another player before the real cuts are made.

Squeezing procedure. A referee moves a knife from left to right across a cake. The

players are instructed to call “stop” when the knife reaches the 1/3 point for each. Let

the ﬁrst player to call “stop” be player 1. (If two or three players call “stop” at the same

time, randomly choose one.) Have player 1 place a mark at the point where she calls

“stop” (the right boundary of piece A in the diagram), and a second mark to the right

that bisects the remainder of the cake (the right boundary of piece B). Thereby player

1 indicates the two points that, for her, trisect the cake into pieces A, B, and C, which

will be assigned after possible modiﬁcations.

ABC

/---------- | ---------- | ----------/

Because neither player 2 nor player 3 called “stop” before player 1 did, each of players

2 and 3 thinks that piece A is at most 1/3. They are then asked whether they prefer

piece B or piece C. There are three cases to consider:

1.Ifplayers2 and 3 each prefer a diﬀerent piece—one player prefers piece B and

the other piece C—we are done: Players 1, 2,and3 can each be assigned a piece

that they consider to be at least tied for largest.

2. Assume players 2 and 3 both prefer piece B. A referee places a knife at the right

boundary of B and moves it to the left. At the same time, player 1 places a knife

at the left boundary of B and moves it to the right in such a way that the value

of cake traversed on the left (by B’s knife) and right (by the referee’s knife) are

equal for player 1. Thereby pieces A and C increase equally in player 1’s eyes.

At some point, piece B will be diminished suﬃciently to a new piece, labeled



—in either player 2’s or player 3’s eyes—to tie with either piece A



or C



,the

enlarged A and C pieces. Assume player 2 is the ﬁrst, or tied for the ﬁrst, to call

“stop” when this happens; then give player 3 piece B



, which she still thinks is

the most valued or the tied-for-most-valued piece. Give player 2 the piece he

thinks ties for most valued with piece B



(say, piece A



), and give player 1 the

remaining piece (piece C



), which she thinks ties for most valued with the other

enlarged piece (A



). Clearly, each player will think he or she received at least a

tied-for-most-valued piece.

⁴ See Brams and Kilgour 2001; Haake, Raith, and Su 2002; Potthoﬀ2002;andAbdulKadiroglu,

Sönmez, and Unver 2004 for other approaches, based on bidding, to the housemates problem discussed

in Su 1999. On approximate solutions to envy-freeness, see Zeng 2000.

steven j. brams 429

3. Assume players 2 and 3 both prefer piece C. A referee places a knife at the

right boundary of B and moves it to the right. Meanwhile, player 1 places a knife

at the left boundary of B and moves it to the right in such a way as to maintain

the equality, in her view, of pieces A and B as they increase. At some point, piece

C will be diminished suﬃciently to C



—in either player 2’s or player 3’s eyes—to

tie with either piece A



or B



, the enlarged A and B pieces. Assume player 2 is

the ﬁrst, or the tied for ﬁrst, to call “stop” when this happens; then give player

3 piece C



, which she still thinks is the most valued or the tied-for-most-valued

piece. Give player 2 the piece he thinks ties for most valued with piece C



(say,

piece A



), and give player 1 the remaining piece (piece B



), which she thinks ties

for most valued with the other enlarged piece (A



). Clearly, each player will think

he or she received at least a tied-for-most-valued piece.

Note that who moves a knife or knives varies, depending on what stage is reached

in the procedure. In the beginning, I assume a referee moves a single knife, and the

ﬁrst player to call “stop” (player 1) then trisects the cake. But, at the next stage of

the procedure, in cases (2)and(3), it is a referee and player 1 that move two knives

simultaneously, “squeezing” what players 2 and 3 consider to be the most valued piece

until it eventually ties, for one of them, with one of the two other pieces.

3 Several Divisible Goods

.............................................................................

Most disputes—divorce, labor–management, merger–acquisition, and inter-

national—involve only two parties, but they frequently involve several homogeneous

goods that must be divided, or several issues that must be resolved.

As an example

of the latter, consider an executive negotiating an employment contract with a

company. The issues before them are (1) bonus on signing, (2) salary, (3)stock

options, (4) title and responsibilities, (5) performance incentives, and (6)severance

pay (Brams and Taylor 1999a).

The procedure I describe next, called adjusted winner (AW), is a two-player pro-

cedure that has been applied to disputes ranging from interpersonal to international

(Brams and Taylor 1996, 1999b).

It works as follows. Two parties in a dispute, after

perhaps long and arduous bargaining, reach agreement on (i) what issues need to

be settled and (ii) what winning and losing means for each side on each issue. For

example, if the executive wins on the bonus, it will presumably be some amount that

the company considers too high but, nonetheless, is willing to pay. On the other hand,

if the executive loses on the bonus, the reverse will hold.

⁵ Dividing several homogeneous goods is very diﬀerent from cake-cutting. Cake-cutting is most

applicable to a problem like land division, in which hills, dales, ponds, and trees form an incongruous

mix, making it impossible to give all of one thing (e.g. trees) to one player. By contrast, in property

division it is possible to give all of one good to one player. Under certain conditions, two-player cake

division, and the procedure to be discussed next (adjusted winner), are equivalent (Jones 2002).

⁶ Procedures applicable to more than two players are discussed in Young 1994; Brams and Taylor 1996,

1999b; Moulin 2003.

430 fair division

Thus, instead of trying to negotiate a speciﬁc compromise on the bonus, the

company and the executive negotiate upper and lower bounds, the lower one favoring

the company and the upper one favoring the executive. The same holds true on other

issues being decided, including non-monetary ones like title and responsibilities.

Under AW, each side will always win on some issues. Moreover, the procedure

guarantees that both the company and the executive will get at least 50 per cent of

what they desire, and often considerably more.

To implement AW, each side secretly distributes 100 points across the issues in a

dispute according to the importance it attaches to winning on each. For example,

suppose the company and the executive distribute their points as follows, illustrating

that the company cares more about the bonus than the executive (it would be a bad

precedent for the company to go too high), whereas the reverse is true for severance

pay (the executive wants to have a cushion in the event of being ﬁred):

Issues Company Executive

1.Bonus 10 5

2. Salary 35 40

3.StockOptions 15 20

4. Title and Responsibilities 15 10

5. Performance Incentives 15 5

6.SeverancePay 10 20

Total 100 100

The boldface ﬁgures show the side that wins initially on each issue by placing more

points on it. Notice that whereas the company wins a total of 10 + 15 + 15 = 40 of its

points, the executive wins a whopping 40 + 20 + 20 = 80 of its points.

This outcome is obviously unfair to the company. Hence, a so-called equitabil-

ity adjustment is necessary to equalize the points of the two sides. This adjust-

ment transfers points from the initial winner (the executive) to the loser (the

company).

The key to the success of AW—in terms of a mathematical guarantee that no win-

win potential is lost—is to make the transfer in a certain order (for a proof, see

Brams and Taylor 1996, 85–94). That is, of the issues initially won by the executive,

look for the one on which the two sides are in closest agreement, as measured by

the quotient of the winner’s points to the loser’s points. Because the winner-to-loser

quotient on the issue of salary is 40/35 = 1.14, and this is smaller than on any other

issue on which the executive wins (the next-smallest quotient is 20/15 = 1.33 on stock

options), some of this issue must be transferred to the company.

But how much? The point totals of the company and the executive will be equal

when the company’s winning points on issues 1, 4,and5,plusx per cent of its points

on salary (left side of equation below), equal the executive’s winning points on issues

2, 3,and6,minusx per cent of its points on salary (right side of equation):

40 + 35x =80−40x

75x =40.

steven j. brams 431

Solving for x gives x =8/15 = 0.533. This means that the executive will win about

53 per cent on salary, and the company will lose about 53 per cent (i.e. win about

47 per cent), which is almost a 50–50 compromise between the low and high ﬁgures

they negotiated earlier, only slightly favoring the executive.

This compromise ensures that both the company and the executive will end up

with exactly the same total number of points after the equitability adjustment:

40 + 35(.533) = 80 − 40(.533) = 58.7.

On all other issues, either the company or the executive gets its way completely (and

its winning points), as it should since it valued these issues more than the other side.

Thus, AW is essentially a winner-take-all procedure, except on the one issue on

which the two sides are closest and which, therefore, is the one subject to the eq-

uitability adjustment. On this issue a split will be necessary, which will be easier if

the issue is a quantitative one, like salary, than a more qualitative one like title and

responsibilities.

Still, it should be possible to reach a compromise on an issue like title and respon-

sibilities that reﬂects the percentages the relative winner and relative loser receive

(53 per cent and 47 per cent in the example). This is certainly easier than trying to

reach a compromise on each and every issue, which is also less eﬃcient than resolving

them all at once according to AW.

In the example, each side ends up with, in toto, almost 59 per cent of what it desires,

which will surely foster greater satisfaction than would a 50–50 split down the middle

on each issue. In fact, assuming the two sides are truthful, there is no better split for

both, which makes the AW settlement eﬃcient.

In addition, it is equitable, because each side gets exactly the same amount above

50 per cent, with this ﬁgure increasing the greater the diﬀerences in the two sides’

valuations of the issues. In eﬀect, AW makes optimal trade-oﬀsbyawardingissues

to the side that most values them, except as modiﬁed by the equitability adjustment

that ensures that both sides do equally well (in their own subjective terms, which

may not be monetary). On the other hand, if the two sides have unequal claims or

entitlements—as speciﬁed, for example, in a contract—AW can be modiﬁed to give

each side shares of the total proportional to its speciﬁed claims.

Can AW be manipulated to beneﬁt one side? It turns out that exploitation of the

procedure by one side is practically impossible unless that side knows exactly how

the other side will allocate its points. In the absence of such information, attempts at

manipulation can backﬁre miserably, with the manipulator ending up with less than

the minimum 50 points its honesty guarantees it (Brams and Taylor 1996, 1999b).

While AW oﬀers a compelling resolution to a multi-issue dispute, it requires careful

thought to delineate what the issues being divided are, and tough bargaining to

⁷ AW may require the transfer of more than one issue, but at most one issue must be divided in the

end.

⁸ Aprocedurecalledproportional allocation (PA) awards issues to the players in proportion to the

points they allocate to them. While ineﬃcient, PA is less vulnerable to strategic manipulation than AW,

with which it can be combined (Brams and Taylor 1996, 75–80).